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I don't think that's true. As far as I see, you can't represent the last digit of pi at all in unary. You can represent the positive integers in unary. I suppose you could represent the negative integers. You cannot represent zero. You can represent real numbers as a fraction, but as far as I know, not as a decimal (sic).

Actually there is an algorithm to compute the n-th digit of Pi without computing the rest.

Okay, so what's the last digit of Pi?

We here in Flatland always get a hearty chuckle when we read about human antics. As you know, we are a Base-pi society as opposed to basing our number system on something as arbitrary as the number of digits we have (btw, what's a digit again?).

I was thinking, you should ask them for Pi to 10 trillion decimal places... but then I thought, by the time they sent you the first half of all those text messages (something like ~31 billion assuming 161 characters max), they would have enough time to calculate the next 5 trillion, along with making a crapload of money from all the fees.

I've heard that in the book (not movie) "Contact" that when Jodie Foster's character meets the uber-aliens she asks them:

"Do you believe in God?"-"Yes"Taken aback "Really, why?"-"We have proof, when PI is expended out to (some number), there is a message"...

I really wish I read the book to know what the message is (maybe "Nietsche is dead"?)

I no longer login because I feel that while attacking a company's products is fair game (specifically Apple), having stories singling out their users as "selfish" and unkind is not "news for nerds stuff that matters". Am I an Apple fanboi? Let's just say I've used NIX for decades (yes I'm old) and I'm not talking OS X.

The aliens are vague about the location of the message (it might be in pi) so the Foster character runs software to search for it. Right at the end of the book her program finds a pattern (A circle drawn in 1s and 0s in an 11 by 11 matrix). This pulls together the thread in the book about belief in god vs religion. It turns out that somebody made the universe after all, and the Christians had been (sort of) right all along, though the scientists were right to demand evidence.

I love both the book and film. Thats unusual for me. The Postman was a fantastic book. Don't get me started on the movie.

I often put the DVD of Contact on just to watch the sequence where Fosters character first hears the signal and her crew reconfigure the telescope to analyse it. Its a classic tech scene.

Much agreed. I like the book a lot better than the movie (the characterizations are superb), but its perhaps one of the best big screen adaptations i've seen, and a gripping sci-fi movie for people who usually don't enjoy sci-fi.

I always found that concept, encoding a message in pi, to be staggeringly stupid. The value of pi doesn't depend on physics, which is why we are able to determine it algorithmically rather than experimentally. (Some people argue 'but pi is the ratio of the circumference of a circle to its diameter, and that depends on physics'. Yes, that ratio depends on physics, for physical circles, provided that some other physical geometry besides 'flat' is possible. But a non-flat geometry would just mean that the

But maybe that just demonstrates the limits of our thinking. We re used to the parameters of our universe and have trouble imagining how things could be different.

Of course there are many, many, marvelous things that are beyond our imagination. But the abstraction pi can be understood to be what it is, within the framework that defines it. This isn't affected by those other unknown things. If there were a 'different' pi that could be conceived of in some other realm, it would have other properties and relationships, and could be given a different name to distinguish it from the one we work with. It is not physical, is not measured, is not a 'parameter of our univ

I think that it's one of those big things that we just don't quite can wrap our minds around. What does it mean that there's a certain structure and properties to geometries, that topology of things looks just like so and not some other way, etc? Do those things depend at all on the universe we live in? Would, somehow, someone in another universe find, that PI has a different value even though it's not a physical constant as far as we can tell? And maybe we're just mistaken and it is really a physical const

I'm serious. How would you even start an argument about PI not being a physical constant? It's really just a matter of definition, and in that sense there's no argument.

But we say that physical constants are some things we measure, and other seemingly fundamental things we can measure are not (like PI). PI can be of course measured to a good few digits by manufacturing a sphere or a disk/cylinder, and then measuring the circumference and radius. We then also have mathematical theories that can come up with

The aliens are vague about the location of the message (it might be in pi) so the Foster character runs software to search for it. Right at the end of the book her program finds a pattern (A circle drawn in 1s and 0s in an 11 by 11 matrix). This pulls together the thread in the book about belief in god vs religion. It turns out that somebody made the universe after all, and the Christians had been (sort of) right all along, though the scientists were right to demand evidence.

If you're given a free hand at the decryption code, you can find any message you want. Presumably the infinite non-repeating sequence of digits is full of marvelous patterns when displayed on a grid as well.

Maybe *every* pattern on every grid size, but I'm not sure of that. (The digits aren't actually independent random numbers.)

It's just a matter of time until some charlatan claims to find a message in our DNA. In a society that can't grok what's the deal with The Bible Codes, people will believe him.

If you're given a free hand at the decryption code, you can find any message you want... Maybe *every* pattern on every grid size"

Yes, given infinite digits, every pattern would appear eventually. However, the point that was made in the book was that the probability of a particular pattern appearing is vanishingly small. In the book Contact, the embedded circle of 1's in and 11 x 11 grid appeared after a LONG (>10^6) sequence of just 0's... and followed by one too. Then PI continued as always. As

The problem is, if you look long enough, hard enough, any message you can think of will appear in pi. Put it in Base 26 and you'll eventually find the complete works of Shakespeare (it might be 10^10^10^10 digits down, but it will be there). I was kind of disappointing in the book that Sagan didn't at least discuss the probability of finding something that appears significant by the time they reached the depth they were at.

I no longer login because I was modded down to terrible karma when I tried to stand up for one of Apple's gay products, and subsequently bragged about performing fellatio on Steve Jobs. People thought I was trolling but actually I was telling the truth.. Am I an Apple fanboi? Yes Indeed.

Actually, it's quite safe to calculate Pi in binary, if you do enough of it. After all, somewhere in it you'll find a message from each copyright owner, signed with his secret key, that you are allowed to have a copy of the copyrighted work. Moreover, you'll have documents about everyone on earth which reveal facts they rather would not like to be published. So actually having enough digits of Pi in binary gives you near-absolute power! That's why THEY want to scare you away from calculating Pi in binary.

"We have proof, when PI is expended out to (some number), there is a message"...

Of course, pi is normal [wikipedia.org] in binary. Every possible message will occur eventually. So if we expand pi far enough, we might even find a positive review for Carrot Top's act. Turns out that math can be wrong.

They just took the number 3.14159 and added a load of random digits to the end - let's face it, nobody's going to check!

Reminds me of the MAX light rail station in the zoo tunnel in Portland, Oregon. Apparently there is the first 100 (1000?) digits of pi chiseled into one of the walls. A writer noticed that the first digits were correct, but quickly went astray. But later in the sequence, there was a recognizable early string of digits. The writer sleuthed that the sculptor had used the Book of Pi, which has the numbers in blocks of ten digits in five (or so) columns. In the book, you read the first row and then the next row.* The sculptor had read the first column, then the next column...

Not a lot. Except to prove that your supercomputer is reliable when calculating numbers like that, and how fast it can do it. Usually, I think it's just used as a test of the computer's abilities rather than anything serious.

Even in the precision engineering world, more than about 10 digits of accuracy for pi is a bit silly. Pi will never really, practically, be required in more depth than what your processor's registers can hold.

There are a number of people that assert some meaning will be found in such natural numbers. It's one of the most basic ratios in existence, and more than one piece of fiction has asserted that meaning will be found in the digits. Such things add a curiosity to the number - will it ever end or ever repeat? could there be a message coded in it? But mainly it's a convenient computational benchmark.

Just to be sure, have the sent the digits to the SETI program looking for patterns? There is some talk that beyond some 2 or 3 billion digits there is a message that apparently begins, "O Brhama, I have created Thee to build the universe, You shall create the universe in accordance to these Laws called Vedas...."

Hmm, I'm not I like this. Has anybody considered the security impact of this? Pi being a proper irrational number is bound to have, as substrings of digits in it's decimal representation, all possible combinations of characters represented as eg. UTF-8, so somebody could easily find all passwords currently in use in there, lined up alphabetically. Somebody clearly hasn't thought this through.

The "world's fastest laser printer" prints about 60 ppm. At one page a second, 10,000 digits per page, it would take 500 million seconds or fifteen years to print it out. So one might hope to live to see all the known digits of pi printed out... unless those pesky computer scientists calculate more of them. But, really, 5 trillion digits ought to be enough for anybody.

If you want to prove that all the digits are correct, you only have to check a few things:

1. There is a sound mathematical proof that the algorithm used in fact does generate the digits of pi, and
2. The algorithm was coded correctly. This should be even easier to check, though likely more tedious.

Now, what it's good for is a little harder. There is no physical application for such a highly accurate value of pi (39 digits should be sufficient to calculate the circumference of the known universe given its radius to within the diameter of a hydrogen atom). However, large numbers of digits of pi are useful as arguments in number theory, statistics, and information theory. For instance, there is no real proof that pi is a normal number [wikipedia.org], but as more digits of pi are found and the statistical properties of the digits are analyzed and shown to be consistent with the definition of normal numbers, that makes the conjecture that pi is actually normal a little closer to being true (see experimental mathematics [wikipedia.org]).

Or some kind of wierd, rare CPU bug. (I was going to mention ram bits getting flipped by cosmic rays and not error corrected, but you've basically covered that with the faulty RAM thing). Oh, you could also have a faulty sector on a hard drive/NAS that you are saving the result too. Or maybe a random network error that corrupts the data (if it gets transmitted over any kind of network). Maybe some wierd glitch in the Front Side Bus (or other hardware on the MoBo which interconnects things).

Knowing that the algorithm is correct, the implementation was codec correctly and you don't have faulty RAM that flips a bit doesn't help if your floating point operations decide to round up or down a single bit due to resolution. Checking the Chudnovsky algorithm [wikipedia.org] it's hard for me to tell how to properly perform the ratio for large values of k. That's where "subtle approximation" begins.

For instance, there is no real proof that pi is a normal number, but as more digits of pi are found and the statistical properties of the digits are analyzed and shown to be consistent with the definition of normal numbers, that makes the conjecture that pi is actually normal a little closer to being true

The problem with normality is that every digit, including the infinitely many that we haven't calculated (and the infinitely many that we never will) are equally significant. We are no closer to determining

If you want to prove that all the digits are correct, you only have to check a few things:

1. There is a sound mathematical proof that the algorithm used in fact does generate the digits of pi, and
2. The algorithm was coded correctly. This should be even easier to check, though likely more tedious.

Actually, 1 isn't very hard. It's known that the series expansion used approaches pi in the limit. If you mean each of the algorithms that they use to break down the Chudnovsky formula, then that's harder. 2 is the real kicker along with hardware errors as others have noted. Basically it was not fully verified that the coding was done correctly. How many things really have mathematically proven and truly 0 bug coding anyway? I don't think even medical or nuclear installations have that.

If you want to prove that all the digits are correct, you only have to check a few things:
1. There is a sound mathematical proof that the algorithm used in fact does generate the digits of pi, and
2. The algorithm was coded correctly. This should be even easier to check, though likely more tedious.

No, that isn't practical or sufficient, and it's not how they actually did it [numberworld.org]. Proving nontrivial pieces of software to be correct is basically impossible, and really you'd also need to prove that the compiler

First off, it's a simple test. A proving ground of sorts. It's also a good place for a programmer to cut his teeth on a lot of concept that he can relate to with other programmers since it is so wide spread.

But don't we have algorithms which let us calculate pi to an arbitrary number of digits?

Yes, we do. Mathematical algorithms, i.e., equations on paper.

Well-known series methods computed using algorithms which have been tuned and re-tuned to the point where it's not really possible to make further major computational optimizations?

Absolutely not. The algorithms have to run on practical, exists-on-the-Earth-today computers. Try to multiply two, million-digit numbers together on your laptop and you'll see what I mean. These achievements are all about computational optimizations. RTFA -- especially the sections entitled "Arithmetic Algorithms" and "Maximizing Scalability." Even the algorithm used for multiplication changes (dynamically!) during the program's execution, based on the size of the operands.

Therefore this isn't so much a new accomplishment as it is "hey look, I left my pi calculating program running longer than the last guy" modified by the occasional minor optimization tweak and running on faster hardware?

Not even close. The computations are so long, and so intense, that errors caused by hardware imperfections can be expected, so error detection and correction algorithms have to be added. If "I left my pi calculating program running longer than the last guy" it would not produce the correct result -- even if the data structures and algorithms it used were up to the task.

But is it really, really something that's newsworthy?

In a word, yes. Could you do it? It's a very, very difficult technical feat, one that required hardware powers and software abilities far beyond those of mortal men. Besides, you're worried about newsworthiness when the two previous/. articles are on wall-climbing robots and the popularity of video game arcades in New York?

And if hypothetical "needing pi to 5 trillion digits" guy needed it to that precision that badly - wouldn't he have already let the calculation run long enough to get it already if this particular calculation only took 90 days?

This isn't about needing pi to 5 trillion digits. This is about learning how to do large computations faster. Like, improving the state of the art.

Either used fixed-point (yuck), symbolic calculations and then only finding the decimal expansion at the last stage, or rewrite your formula to avoid any possible lack of precision (i.e. any division).